It may be noted that \(F_{\pm }\)  is necessarily hermitian, while \(K(\xi _1, \xi _2)\)  is hermitian either for (i) \(\xi _1=\xi _2\)  or (ii) \(\xi _1 \ne \xi _2, T_+ = -T_-\) . The condition \(T_+=-T_-\)  is achieved for \(\Gamma \mu (t)=0\) .
The choice \(\mu (t)=0 \ \forall \ t\)  is excluded, since LC as well as BLG terms also vanish
in this limit. Thus, for \(\xi _1 \ne \xi _2\) , \(K(\xi _1, \xi _2)\)  is hermitian for \(\Gamma = 0\) . i.e. the limit of
vanishing loss-gain terms. In general, the nonlinear term in Eq. (REF ) is non-hermitian —the non-unitary transformation involving \(U(t)\)  removes the loss-gain terms at the cost of introducing
non-hermiticity in the non-linear term. The nonlinear term in Eq. (REF ) becomes hermitian
either for (i) \( g_{11}=g_{21}, g_{22}=g_{12}, \Gamma \ne 0\)  or (ii)\(\Gamma = 0\) ,
since both \(K(g_{11}, g_{21})\)  and \(K(g_{11}, g_{21})\)  are hermitian in these limits. The second condition
corresponds to vanishing loss-gain term for which \(U(t)\)  is unitary and the result is expected.
The nonlinear term for the first condition takes the form of MZS system[1]}, [2]}
\(\left[ \Psi ^{\dagger } \left( g_{11} P_+ + g_{22} P_- \right) \Psi \right] \Psi \) , which remains
real-valued after the non-unitary transformation. This particular form of nonlinear interaction
with \(V(x)=0, \mu (t)=1\)  and space-time independent \(g_{11}, g_{22}\)  has been investigated earlier[3]}.
